*2.2. Flow Analysis*

Using boundary layer approximations, the governing equations involving the magnetic effect on Sisko blood flow with gold nanoparticles over a porous, curved surface with radiation and partial slip are

$$\frac{\partial}{\partial r}[(r+R)v] + R\frac{\partial u}{\partial s} = 0,\tag{1}$$

$$\frac{1}{\left(r+R\right)}\mu^2 = \frac{1}{\rho\_{\rm nf}}\frac{\partial p}{\partial r},\tag{2}$$

$$\begin{split} \frac{\partial \frac{\partial u}{\partial r}}{\partial r} + \frac{b}{\rho\_{nf}(r+R)^2} \frac{\partial}{\partial r} \left[ (r+R)^2 \left( -\left(\frac{\partial u}{\partial r} - \frac{u}{(r+R)}\right) \right)^n \right] + \left( \frac{Ru}{r+R} \right) \frac{\partial u}{\partial s} + \\ \frac{u\upsilon}{(r+R)} = -\frac{1}{\rho\_{nf}} \left( \frac{R}{r+R} \right) \frac{\partial p}{\partial s} - \frac{\sigma\_{nf} R\_0^2}{\rho\_{nf}} u + \frac{\mu\_{nf}}{\rho\_{nf}(r+R)^2} \frac{\partial}{\partial r} \left[ (r+R)^2 \left( \frac{\partial u}{\partial r} - \frac{u}{(r+R)} \right) \right], \end{split} \tag{3}$$

$$\frac{\partial T}{\partial r} + \frac{1}{\left(c\_p \rho\right)\_{nf} (r+R)} \frac{\partial}{\partial r} [(r+R)q\_r^\*] + \left(\frac{Ru}{r+R}\right) \frac{\partial T}{\partial s} = \frac{k\_{nf}}{\left(c\_p \rho\right)\_{nf}} \left(\frac{\partial^2 T}{\partial r^2} + \frac{1}{\left(r+R\right)} \frac{\partial T}{\partial r}\right), \tag{4}$$

with the boundary conditions

$$\begin{cases} \ u = \lambda \mathcal{U}\_{\overline{w}} + L \left[ \frac{\partial \underline{u}}{\partial r} - \frac{\underline{u}}{r + R} \right], \ v = v\_{0\prime} \ T = T\_{\overline{w}} \text{ at } r = 0, \\\ u \to 0, \ \frac{\partial \underline{u}}{\partial r} \to 0, \ T \to T\_{\infty} \text{ as } r \to \infty. \end{cases} \tag{5}$$

Here, *u* and *v* are components of the velocity, such that the corresponding stretching velocity moves along the axial *s* direction and suction along the radial *r* direction, respectively. In addition, *n* and *b* are material constants, *L* constant slip parameter, *p* the pressure, *R* the radius of curvature, and *T* the nanofluid temperature. In Equation (4), the relative heat flux is represented by *q*∗*r* and can be articulated by employing the Rosseland approximation as

$$q\_r^\* = -4(3k\_1)^{-1} \sigma\_1 \frac{\partial T^4}{\partial r} \,\,\,\,\,\,\tag{6}$$

where *σ*1 is the mean proportion coefficient and *k*1 the Stefan–Boltzmann constant. Thus, the term *T*<sup>4</sup> at point *T*∞ is exercised by the Taylor series. Avoiding the highest-order terms, we ge<sup>t</sup>

$$T^4 \approx 4TT\_{\infty}^3 - 3T\_{\infty}^4\tag{7}$$

The thermo-physical quantities of the gold particle nanofluid introduced in the governing equations are given by

$$\begin{aligned} \frac{\mu\_{nf}}{\mu\_f} &= 1 + 7.3\phi + 123\phi^2 \quad \text{for} \quad \phi > 0.02, \quad \frac{\rho\_{nf}}{\rho\_f} = (1 - \phi) + \frac{\rho\_{sf}}{\rho\_f}\phi, \\\frac{\left(\varepsilon\_f \rho\right)\_{nf}}{\left(\varepsilon\_p \rho\right)\_f} &= (1 - \phi) + \frac{\left(\varepsilon\_f \rho\right)\_{s\_1}}{\left(\varepsilon\_p \rho\right)\_f}\phi, \quad \frac{k\_{nf}}{k\_f} = \left[\frac{\left(k\_{s\_1} + 2k\_f\right) - 2\left(\phi k\_f - \phi k\_{s\_1}\right)}{\left(k\_{s\_1} + 2k\_f\right) + \phi\left(k\_f - k\_{s\_1}\right)}\right], \\\frac{\sigma\_{nf}}{\sigma\_f} &= \left(1 + \frac{3\phi\left(\frac{\sigma\_{s\_1}}{\sigma\_f} - 1\right)}{\left(\frac{\sigma\_{s\_1}}{\sigma\_f} + 2\left(-\frac{\sigma\_{s\_1}}{\sigma\_f} - 1\right)\phi}\right), \end{aligned} \tag{8}$$

where *k f* , *ρf* , *μf* and *σf* are the thermal conductivity, density, viscosity, and electrical conductivity of the carrier-based fluid, respectively; *kn f* , *ρn f* , *μn f* and *σn f* are the corresponding quantities of the nanofluid, respectively; *cp* is the heat capacity; and subscripts *f* , *n f* , and *s*1 are quantities of the carrier-based fluid, the nanofluid, and the solid volume fraction of the nanoparticles, respectively.

Upon applying the following similarity transformation:

 $\mu = \text{cs}F'(\eta), \text{ } \upsilon = \frac{-\text{cs}R}{r+R}\text{Re}\_b^{-\frac{1}{n+1}}\frac{1}{n+1}[2nF(\eta) + (1-n)\eta F'(\eta)],$  
$$\psi = \text{cs}^2\text{Re}\_b^{-\frac{1}{n+1}}F(\eta), \eta = \left(\frac{r}{\delta}\right)\text{Re}\_b^{\frac{1}{n+1}}, \eta = \rho\_f\text{c}^2\text{s}^2P(\eta), \theta(\eta) = \frac{T-T\_\infty}{T\_w-T\_\infty}.$$

Equation (1) is satisfied identically, whereas Equations (2)–(5) become

$$\frac{\frac{\partial P}{\partial \eta}}{\frac{\partial \eta\_f}{\partial f}} = \frac{F'^2}{\eta + B} \tag{10}$$

$$\begin{split} \frac{2RP}{\left(\eta+B\right)\frac{\nu\_{sf}}{\vartheta\_{f}}} &= \frac{\mathcal{B}}{\eta+B} \left(\frac{2u}{n+1}\right) \left(FF'' + \frac{FF'}{\eta+B}\right) - \frac{\mathcal{B}}{\eta+B}F'^2 + \frac{\frac{\mathcal{F}\_{uf}}{\mathcal{F}\_{f}}}{\frac{\mathcal{F}\_{ff}}{\mathcal{F}\_{f}}} B\_{1} \left(F'' \frac{F'}{\left(\eta+B\right)^{2}} + \frac{F''}{\eta+B}\right) + \\ \frac{\mathcal{B}}{\frac{\mathcal{B}}{\eta+B}} &- \left(F' \frac{F'}{\eta+B}\right) \left(F''' + \frac{F'}{\left(\eta+B\right)^{2}} - \frac{F''}{\eta+B}\right) - \frac{2}{\left(\eta+B\right)\frac{\nu\_{sf}}{\vartheta\_{f}}} \left(-\left(F''\frac{F'}{\eta+B}\right)\right)^{n} - M\frac{\frac{\mathcal{F}\_{uf}}{\eta+B}}{\frac{\mathcal{F}\_{ff}}{\eta\_{f}}} F', \end{split} \tag{11}$$

By removing the pressure term from Equations (10) and (11), we obtain the following equations, along with the dimensionless form of the energy equation:

Σ1 Σ2 *<sup>B</sup>*1*<sup>F</sup>* + 2*F η*+*B* + *F* (*η*+*<sup>B</sup>*)<sup>3</sup> − *F* (*η*+*<sup>B</sup>*)<sup>2</sup> + 2*n n*+1 ⎡ ⎢⎢⎢⎢⎢⎣ *Bη*+*B* (*FF* + *<sup>F</sup>F*)+ *B* (*η*+*<sup>B</sup>*)<sup>2</sup> *FF* + *F*2 − *BFF* (*η*+*<sup>B</sup>*)<sup>3</sup> ⎤ ⎥⎥⎥⎥⎥⎦ − 2*BFF η*+*B* − 2*BF*<sup>2</sup> (*η*+*<sup>B</sup>*)<sup>2</sup> + *n*Σ2 −*<sup>F</sup> F η*+*B n*−1⎛⎜⎝ *F* + 2*F η*+*B* − *F* (*η*+*<sup>B</sup>*)<sup>2</sup> + *F* (*η*+*<sup>B</sup>*)<sup>3</sup> ⎞⎟⎠− *<sup>n</sup>*(*<sup>n</sup>*−<sup>1</sup>) Σ2 −*<sup>F</sup> F η*+*B <sup>n</sup>*−<sup>2</sup>*<sup>F</sup>* + *F* (*η*+*<sup>B</sup>*)<sup>2</sup> − *F η*+*<sup>B</sup>* <sup>2</sup> − Σ3 Σ2 *M F η*+*B* + *F* = 0, (12) Σ4 + 43*Rd <sup>θ</sup>* + *θ η* + *B* + PrΣ5 *B η* + *B* 2*n n* + 1 *<sup>F</sup>θ* = 0, (13)

in which:

$$\frac{\mu\_{nf}}{\mu\_f} = \Sigma\_{1\prime} \frac{\rho\_{nf}}{\rho\_f} = \Sigma\_{2\prime} \frac{\sigma\_{nf}}{\sigma\_f} = \Sigma\_{3\prime} \frac{k\_{nf}}{k\_f} = \Sigma\_{4\prime} \frac{\left(\rho c\_p\right)\_{nf}}{\left(\rho c\_p\right)\_f} = \Sigma\_5.$$
